Nonlinear spin waves for the Heisenberg model and the ferromagnetic-antiferromagnetic bifurcations

We present finite-amplitude spin-wave solutions for the nonlinear equations of the classical Heisenberg model on a general periodic lattice. These are families of periodic solutions bifurcating from the ferromagnetic (FR) and antiferromagnetic (AF or Neel) states which are fixed points of the flow. We allow for longitudinal anisotropy and constant uniform magnetic field in the direction of anisotropy. We find analytical expressions for the energy-frequency (e-w) curves of these families. In particular, we show that the AF families come in pairs of vertical lines. In the expression of both the FR and AF nonlinear spin waves (NLSWs), we are able to eliminate the amplitude in favour of the frequency and/or energy. All special cases of solutions are carefully analysed. We discover that the two AF families end on the FR family of a corresponding wavevector. This correspondence can be made precise without compromising the generality of the lattice. Our major point is the proof that these FR-AF intersections on the e-w plot are isochronous branchings. Hence, we establish a novel view of the AF NLSWs as bifurcations of the FR NLSWs.